How do you add the Jiggly Armature code to Blender as an add-on?
Addons are all just code, in one or more files, normally packaged up in zip archive. However if all the code is in one file, they don't need to be zipped up.

If you go to: https://github.com/cheece/JiggleArmature/blob/master/JiggleArmature.py and right click on the button in the top right that says raw, and select save link as you will download a file containing the addon code. 

You can then install this like a normal addon. Bear in mind this hasn't been updated in 2 years, so it may have bugs. The addon metadata says it was created for 2.77. That said it will probably be ok in 2.79.

As for any side affects from the addon code, you would have to read the code and check it carefully, as with any other addon. I had a very quick skim and it looks like it just does what it says.